↑

Properties

Label 2-325-13.9-c1-0-14
Degree $2$
Conductor $325$
Sign $0.252 + 0.967i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.651 − 1.12i)2-s + (0.5 − 0.866i)3-s + (0.151 + 0.262i)4-s + (−0.651 − 1.12i)6-s + (−0.5 − 0.866i)7-s + 3·8-s + (1 + 1.73i)9-s + (2.80 − 4.85i)11-s + 0.302·12-s − 3.60·13-s − 1.30·14-s + (1.65 − 2.86i)16-s + (0.197 + 0.341i)17-s + 2.60·18-s + (−0.802 − 1.39i)19-s + ⋯
L(s)  = 1  + (0.460 − 0.797i)2-s + (0.288 − 0.499i)3-s + (0.0756 + 0.131i)4-s + (−0.265 − 0.460i)6-s + (−0.188 − 0.327i)7-s + 1.06·8-s + (0.333 + 0.577i)9-s + (0.845 − 1.46i)11-s + 0.0874·12-s − 1.00·13-s − 0.348·14-s + (0.412 − 0.715i)16-s + (0.0478 + 0.0828i)17-s + 0.614·18-s + (−0.184 − 0.318i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57522 - 1.21675i\)
\(L(\frac12)\) \(\approx\) \(1.57522 - 1.21675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + 3.60T \)
good2 \( 1 + (-0.651 + 1.12i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.80 + 4.85i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.197 - 0.341i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.802 + 1.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.10 - 7.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1.80 - 3.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.10 - 3.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.21T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (5.40 + 9.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.40 - 14.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 9.21T + 79T^{2} \)
83 \( 1 + 5.21T + 83T^{2} \)
89 \( 1 + (4.10 - 7.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.80 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.39542642490891031770567349471, −10.84736996503307230754316620310, −9.731794915013741992391298648287, −8.510092599688663820859761106275, −7.54827420226115694303041201209, −6.74359262157074900666196594213, −5.20896297522348838823106162448, −3.92016790893572744347527375906, −2.91937716854347224178348248063, −1.55780284555723357196890941138, 2.05415145370801478712475317421, 3.98071201890444720609780830678, 4.73553918528322940323808604370, 5.98161892275181840749330747399, 6.92339586144494068228484129047, 7.66758887332794813686196299996, 9.240789475778321231571414459340, 9.744307546039387493967920618624, 10.67954763191145735852078674008, 12.12995248868432001520066618736

Graph of the $Z$-function along the critical line